Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

July 24, 2005

Streetfest Workshop 3

Posted by Guest

Let’s start with Katzarkov’s talk on ‘Mirror symmetry and manifolds of general type’. He began with a question: are there instances of distinct (not symplectomorphic) 4D symplectic simply connected mannifolds X1 and X2 such that their Gromov-Witten invariants are equal? It turns out there is. Katzarkov mentioned an example of Donaldson’s, but more generally there is a conjecture that the image of ChK0(Db(W,f)) in ⊕Hi(F) is the sublattice of H2(X,Z) of vanishing cocycles.

So what is all the terminology here? (X,ω) is always a symplectic manifold. F is a perverse sheaf of vanishing cycles for the mirror W. For an f:W→C the categorical equivalence is between the so-called Fukaya category for X and the bounded sheaves Db(W,f) as discussed in the first talk in Sydney.

Other workshop talks…..

After Katzarkov, Getzler dazzled us all once again with open and closed modular operads. Let C be a symmetric monoidal category and P(g,n) a sequence of objects labelled by a genus g and n the number of legs at a vertex. The overall genus of a graph is given by the sum of the vertex genuses plus the first Betti number of the graph. Thus a graph with non-negative integers assigned to each vertex yields composition rules, such as P(1,3)⊗P(0,4)⊗P(3,3)⊗P(2,3)→P(8,3) where 8 is the genus of a graph that the reader may easily identify.

Example: P(g,n) = the moduli space of conformal structures for a Riemann surface of genus g with n punctures. This motivates throwing away graphs and using ‘thicked lines’ or rather surfaces. String theory! The nice category here has oriented surfaces as objects. Morphisms are pairs of surfaces together with a configuration of curves in the target and an isomorphism between the source and the surface obtained from the target by cutting along the curves.

By definition, a modular operad is a symmetric monoidal functor from this category to C.

A slightly different version gives a characterisation of open conformal field theory. We can talk about the Deligne-Mumford compactification of M(g,n;h,m) where as before n is the number of boundary components that are circles and n is the number of boundary components that are arcs etc. This moduli space of compact orbifolds with corners has dimension 6g−6+3h+2n+m. Considering, for example, the case (g,n;h,m)=(0,0;1,m) one obtains the associahedron rule!

Finally, Getzler mentioned the theorem that states that the 2-skeleton of each component of this moduli space is connected.

Now I must go and listen to Higson, Getzler and others talk about NCG.
Marni

Posted at July 24, 2005 11:37 PM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/617